A numerical approach to the modeling of Thompson and Troian slip on magnetized flow of Al 2 O 3 – PAO nanolubricant over an inclined rotating disk

Abstract

In automotive fluids, hydraulic, gear, and bearing oils, as well as in applications operating in extremely high or cold temperatures, PAO is widely employed. In present work, we have made an attempt to develop a mathematical model to discuss the flow of magnetized $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant over an inclined rotating disk in Darcy-Forchheimer porous medium with Thompson and Troian slip at the boundary. The effects of mixed convection, nonlinear heat radiation, viscous dissipation, Joule heating, and non-uniform heat source/sink are also included in the modeling. We have solved the proposed model numerically with the help of MATLAB built-in bvp-4c method after metamorphosing the PDEs to ODEs. The enhancing values of inertial parameter and velocity slip parameter decrease the tangential and radial velocities of the nanolubricant. The temperature of the nanolubricant $A{l}_2{O}_3-\mathrm{PAO}$ enhance significantly by strengthening the magnetic field, whereas radial and tangential velocities get retarded. The non-uniform heat source/sink parameters play a vital role in controlling heat transmission phenomenon. The increasing values of Eckert number, radiation parameter, and non-uniform heat generation parameters tend to increase the value of Nusselt number. The value of Nusselt number drops with rising values of Biot number and non-uniform heat sink parameters.

Introduction

Lubricants usually include additives and might be based on mineral, synthetic, or vegetable oils. To lessen friction and wear, they are utilized as low-shear-resistance coatings between affecting surfaces. In addition to enhancing opposition to corrosion temperature and oxidation, the major goal of additives is to boost the performance and longevity of lubricants. Synthetic lubricants which have the ability to maintain their features at high temperatures without decomposing or combusting or experiencing chemical degradation have been developed in recent years in response to the need for high performance lubricants in engineering fields, primarily in space and the automotive industry.^1–6 Polyalphaolefins (PAOs) are among the synthetic lubricants which have gained more attention because of their intrinsic attributes such as wide operating temperature range, thermal stability and better oxidation, low viscosity, biodegradability, lower volatility, and high viscosity indices. PAOs have become new standard as a lubricant; according to the consumer reports, $70\%$ of the vehicles having either PAO or PAO based blended oils. The need for lubricants that efficiently reduce friction and wear under demanding circumstances while being less detrimental to the environment has also been supported by governmental agencies. ⁷ Recent research indicates that nanotechnology can enhance lubricant enactment by lowering friction and system wear. The diffusion of nanoparticles in a base lubricant is known as a “nanolubricant,” which has caught the attention of researchers. Nanoparticles typically have diameters between 1 and 100 nm. The addition of nanoparticles to base oils has the potential to improve certain tribological qualities like friction and wear-resistance. The most significant aspect is that the majority of these particles are environmentally harmless. Despite being a relatively new field of research, nanotechnology has distinguished itself and demonstrated its potential in a number of fields, including tribology. The foremost purposes of using nanoparticles as lubricant additives are to give oil anti-wear qualities, modify friction, and increase resilience to extreme pressure. ⁸ When employed in lubricants, nanoparticles can also facilitate heat transmission and bring down temperatures in the areas where they are used.^9,10 The nanolubricants have been attracting the attention of researchers and scientists due to their tremendous applications. Choi et al. ¹¹ discussed heat dissipation experimentally and theoretically in nanolubricant evaporators. Gamaoun et al. ¹² investigated conduction-convection heat transfer of a fin wetted with nanolubricant $\mathrm{ZnO}-\mathrm{SAE}50$ . Riaz et al. ¹³ discussed the flow of $\mathrm{ZnO}-\mathrm{SAE}50$ nanolubricant over a Riga plate. Nayak et al.^14–16 studied transmission of heat in the flow of $\mathrm{ZnO}-\mathrm{SAE}50$ nanolubricant over various surfaces. Khan ¹⁷ analyzed the consequence of bioconvection in the flow of second grade nanofluid comprising nanoparticles and gyrotactic microorganisms. Zuhra et al. ¹⁸ presented bioconvection’s simulation in the suspension of second grade nanofluid consisting of gyrotactic microorganisms and nanoparticles.

Magneto hydrodynamics (MHD) is the study of magnetic effects and the behavior of electrically conducting fluids. The fundamental theory underlying MHD holds that magnetic fields cause currents to pass through conductive fluids that are in motion. With the use of MHD, researchers have discovered a method for solving some of the issues encountered in engineering. Plasma confinement, nuclear reactor cooling with liquid metal, and electromagnetic casting are a few of these issues. MHD issues can arise in a wide range of circumstances, such as when forecasting space weather, controlling turbulent oscillations in semiconductor melt during crystal development, and measuring beverage flow rates in the food business. The studies^19–27 explain the flow of magnetized fluids over diverse geometries. Khan et al. ²⁸ investigated second grade fluid’s thin film flow over a stretching sheet in porous medium along with heat transfer. The effects of heat source/sink on transfer of heat captured the attention of many researchers. Hsiao ²⁹ analyzed MHD flow of viscoelastic fluid across a stretchable sheet with non-uniform heat source/sink. Ramesh et al. ³⁰ have examined MHD flow across an inclined surface embedded in an incompressible fluid with the effect of non-uniform heat source/sink. Ramandevi et al. ³¹ discussed combined effects of viscous dissipation and non-uniform heat sink/source on MHD flow of the fluid. Sulochana et al. ³² examined the impact of non-uniform heat sink or source in $3D$ flow of Casson fluid. Palwasha et al. ³³ conducted a study to examine thin film fluid’s boundary layer flow having variable properties. Radiative heat, whether linear or nonlinear, has a substantial impact on a range of processes involving extremely high temperatures due to its many industrial and power plant applications. In business, the heat transfer procedure is crucial and necessary for final items to have the right qualities. Modern systems connected to spacecraft, power generator plasma, astrophysical flow, reactor cooling etc. are developed based on radiation applications. Khan et al. ³⁴ discussed the impacts of thermal radiation and thermophoresis with mass and heat transfer in a magnetized thin film flow of second grade fluid having variable properties over a stretching sheet. Some recent articles discussing the effects of nonlinear thermal radiation can be seen in the references.^35–39

A thorough review of the literature reveals that no study is yet done to examine the flow characteristics and heat transference rate in the flow of magnetized $A{l}_2{0}_3-\mathrm{PAO}$ nanolubricant over a rotating inclined disk. The prime motive of this investigation is to fill this gap and introduce $A{l}_2{0}_3-\mathrm{PAO}$ nanolubricant over a rotating inclined disk subjected to Thompson and Troian slip and convective conditions. The flow takes place in Darcy-Forchheimer porous medium. The impacts of mixed convection, nonlinear heat radiation, viscous dissipation, Joule heating, and non-uniform heat source/sink are also included in the modeling. The proposed model is solved numerically via MATLAB built-in bvp-4c method after metamorphosing the PDEs to ODEs. The effects of imperative parameters on concerned profiles have been yakked via graphs.

Mathematical formulation

In current work, a mixed convective flow of magnetized $A{l}_2{0}_3-\mathrm{PAO}$ nanolubricant over a rotating inclined disk in a Darcy Forchheimer porous medium has been considered. At $z=0$ , the disk swirls with uniform angular velocity $\mathrm{\Omega }$ . The components $(u,v,w)$ of velocity are taken along $(r,\varphi ,z)$ directions respectively. In axial direction, a magnetic field having strength $B_0$ is implemented. The problem’s axial symmetry prevents us from including derivatives along $\varphi$ coordinate. The influences of non-uniform heat sink/source, nonlinear heat radiation, viscous dissipation, and Joule heating are considered in the modeling. The flow is subjected to the effects of Thompson and Troian slip and convective conditions. The depiction of problem is made in Figure 1.

Figure 1.

Flow geometry.

Governing equations

The flow phenomenon is expressed in this problem in the following ways^40,41:

$$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,$$

(1)

$$\begin{array}{c}u\frac{\partial u}{\partial r}-\frac{v^2}{r}+w\frac{\partial u}{\partial z}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p}{\partial r}+\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}[\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}+\frac{{\partial }^{2}u}{\partial z^2}]\hfill \\ +\frac{{\left(\rho \beta \right)}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}g(T-T_{\infty })\cos \psi -\frac{{\sigma }_{\mathrm{nf}}{B}_0^{2}u}{{\rho }_{\mathrm{nf}}}-\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}\frac{u}{K^{*}}-F^{*}{u}^2,\hfill \end{array}$$

(2)

$$\begin{array}{c}u\frac{\partial v}{\partial r}+\frac{\mathrm{uv}}{r}+w\frac{\partial v}{\partial z}=\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}[\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^2}+\frac{{\partial }^{2}v}{\partial z^2}]\hfill \\ -\frac{{\left(\rho \beta \right)}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}g(T-T_{\infty })\sin \psi -\frac{{\sigma }_{\mathrm{nf}}{B}_0^{2}v}{{\rho }_{\mathrm{nf}}}-\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}\frac{v}{K^{*}}-F^{*}{v}^2,\hfill \end{array}$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p}{\partial z}+\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}[\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}]}\\ -\frac{{\sigma }_{\mathrm{nf}}{B}_0^{2}w}{{\rho }_{\mathrm{nf}}}-\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}\frac{w}{K^{*}}-F^{*}{w}^2,\hfill \end{array}$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\frac{k_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}[\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}]}\\ +2\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}[{\left(\frac{{\partial }^{2}u}{\partial r^2}\right)}^2+\frac{u^2}{r^2}+{\left(\frac{{\partial }^{2}w}{\partial z^2}\right)}^2]\hfill \\ +\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\left(\rho C_p\right)}_{\mathrm{nf}}}(u^2+v^2)+\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\hfill \\ [{\left(\frac{{\partial }^{2}v}{\partial z^2}\right)}^2+{(\frac{\partial w}{\partial r}+\frac{\partial u}{\partial z})}^2+{\left\{r\frac{\partial }{\partial r}\left(\frac{v}{r}\right)\right\}}^2]\hfill \\ -\frac{1}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{16{\sigma }^{*}{T}_{\infty }^3}{3{\kappa }^{*}}\frac{{\partial }^{2}T}{\partial z^2}+\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{1}{K^{*}}(u^2+v^2)\hfill \\ +\frac{1}{{\left(\rho C_p\right)}_{\mathrm{nf}}}{F}^{*}(u^3+v^3)+\frac{1}{{\left(\rho C_p\right)}_{\mathrm{nf}}}q″\prime .\hfill \end{array}$$

(5)

The conditions at the boundary for present problem are ⁴² :

$$\begin{array}{c}u=\mathrm{rS}+{\gamma }^{*}{(1-{\xi }^{*}\frac{\partial u}{\partial z})}^{-\frac{1}{2}}\frac{\partial u}{\partial z},v=r\mathrm{\Omega }+{(1-{\xi }^{*}\frac{\partial v}{\partial z})}^{-\frac{1}{2}}{\gamma }^{*}\frac{\partial v}{\partial z},w=0,\hfill \\ -k_{\mathrm{nf}}\left(\frac{\partial T}{\partial z}\right)=h_f(T_f-T),\mathrm{at}z=0,\hfill \\ u\to 0,v\to 0,T\to T_{\infty },p\to p_{\infty }\mathrm{as}z\to \infty \hfill \end{array}$$

(6)

The following transformations are required for the current problem ⁴³

$$\begin{array}{c}u=r\mathrm{\Omega }F\prime \left(\eta \right),v=r\mathrm{\Omega }G\left(\eta \right),w=-\sqrt{2\mathrm{\Omega }}{\upsilon }_{\mathrm{SAE}50}F\left(\eta \right),\hfill \\ p=p_{\infty }-\mathrm{\Omega }{\mu }_{f}P\left(\eta \right),\hfill \\ T=T_{\infty }+(T_f-T_{\infty })\theta (\eta ),\eta =\sqrt{\frac{2\mathrm{\Omega }}{{\upsilon }_f}z}.\hfill \end{array}$$

(7)

Where $\eta$ is the non-dimensional distance along rotational axis.

$q″\prime$ in non-dimensional for present situation can be expressed as

$$q″\prime =\frac{k_{\mathrm{nf}}\mathrm{\Omega }}{{\nu }_{\mathrm{nf}}}[A^{*}(T_w-T_{\infty })F\prime +B^{*}(T-T_{\infty })],$$

(8)

where, $B^{*}$ and $A^{*}$ are temperature dependent and space-dependent coefficients. Additionally, $A^{*}>0$ , $B^{*}>0$ represents heat generation state and $A^{*}<0$ , $B^{*}<0$ , correspond to heat absorption of the system.

Equations (2)–(6) take the following form by using equations (7) and (8).

$$\begin{array}{c}{\mathrm{\Omega }}_2(2F″\prime -\mathrm{KF}\prime )+2FF″-{F\prime }^2+G^2+\left(\frac{\mathrm{Gr}}{R{e}^2}\right){\mathrm{\Omega }}_3\left(\cos \mathrm{\Psi }\right)\theta \hfill \\ -\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right){\mathrm{\Omega }}_{1}MF\prime -F_r{F\prime }^2=0,\hfill \end{array}$$

(9)

$$\begin{array}{c}{\mathrm{\Omega }}_2(2G″-\mathrm{KG}\prime )+2FG\prime -F\prime G-\left(\frac{\mathrm{Gr}}{R{e}^2}\right){\mathrm{\Omega }}_3\left(\sin \mathrm{\Psi }\right)\theta \hfill \\ -\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right){\mathrm{\Omega }}_1\mathrm{MG}-F_r{G}^2=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4}{3}\mathrm{Rd}){\mathrm{\Omega }}_4+\mathrm{Pr}[F\theta \prime +6\left(\frac{\mathrm{Ec}}{\mathrm{Re}}\right){\mathrm{\Omega }}_5{F\prime }^2+\mathrm{Ec}{\mathrm{\Omega }}_5({F″}^2+{G\prime }^2)\hfill \\ +\mathrm{Ec}{\mathrm{\Omega }}_{5}K({F\prime }^2+G^2)+\mathrm{Ec}{\mathrm{\Omega }}_4{F}_r({F\prime }^3+G^3)+\frac{1}{2}{\mathrm{\Omega }}_4\hfill \\ \{\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)\mathrm{MEc}({F\prime }^2+G^2)+\frac{1}{{\mathrm{\Omega }}_2}\left(\frac{k_{\mathrm{nf}}}{k_f}\right)\frac{1}{\mathrm{Pr}}(A^{*}F\prime +B^{*}\theta )\}]=0.\hfill \end{array}$$

(11)

The boundary conditions are transformed as

$$\begin{array}{c}F\prime =\gamma +{\gamma }_1{[1-\xi F″\left(0\right)]}^{-\frac{1}{2}}F″\left(0\right),G=1\hfill \\ +{\gamma }_1{[1-\xi G\prime \left(0\right)]}^{-\frac{1}{2}}G\prime \left(0\right),F=0,\hfill \\ \theta \prime =-\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)B_1(1-\theta ,)\mathrm{at}\eta =0\hfill \\ F\prime \to 0,G\to 0,\theta \to 0,\mathrm{as}\eta \to 0,\hfill \end{array}$$

(12)

where

$$\begin{array}{c}K=\frac{{\upsilon }_f}{\gamma K^{*}},\mathrm{Gr}=\frac{g{\beta }_f(T_f-T_{\infty })r^3}{{\upsilon }_f^2},\mathrm{Re}=r\left(\frac{r\mathrm{\Omega }}{{\upsilon }_f}\right),{\gamma }_1={\gamma }^{*}\sqrt{\frac{2\mathrm{\Omega }}{{\upsilon }_f}},\hfill \\ M=\frac{{\sigma }_f{B}_0^2}{\mathrm{\Omega }{\rho }_f},F_r=\frac{C_b}{\sqrt{K^{*}}},\mathrm{Pr}=\frac{{\upsilon }_f}{{\alpha }_f},\mathrm{Rd}=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}{k}_f},\mathrm{Bi}=\frac{h_f}{k_f}\sqrt{\frac{{\upsilon }_f}{2\mathrm{\Omega }}},\hfill \\ \mathrm{Ec}=\frac{r^2{\mathrm{\Omega }}^2}{{\left(C_p\right)}_f(T_f-T_{\infty })},\gamma =\frac{S}{\mathrm{\Omega }},\mathrm{and}\xi ={\xi }^{*}r\mathrm{\Omega }\sqrt{\frac{2\mathrm{\Omega }}{{\upsilon }_f}}\hfill \end{array}$$

We have made the following substitutions

$$\begin{array}{c}{\mathrm{\Omega }}_1=\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}},{\mathrm{\Omega }}_2=\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}},{\mathrm{\Omega }}_3=\frac{{\left(\rho \beta \right)}_{\mathrm{nf}}}{{\left(\rho \beta \right)}_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}},\hfill \\ {\mathrm{\Omega }}_4=\frac{{\left(\rho C_p\right)}_f}{{\left(\rho C_p\right)}_{\mathrm{nf}}},{\mathrm{\Omega }}_5=\frac{{\left(\rho C_p\right)}_f}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}.\hfill \end{array}$$

Physical parameters

The engineering interest quantities such as circumferential wall shear stress ${\tau }_{\varphi }$ and the radial wall stress ${\tau }_r$ , can be expressed as follows with the help of Newtonian formula

At surface, the drag force is represented as

$${\tau }_r={\mu }_f{(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r})}_{z=0}={\mu }_f\left(r\mathrm{\Omega }\right)\sqrt{\frac{2\mathrm{\Omega }}{{\upsilon }_f}}F″\left(0\right),$$

$${\tau }_{\varphi }={\mu }_f{(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial r})}_{z=0}={\mu }_f\left(r\mathrm{\Omega }\right)\sqrt{\frac{2\mathrm{\Omega }}{{\upsilon }_f}}G\prime \left(0\right).$$

The skin friction coefficient is then given as

$$C_f=\frac{\sqrt{{\tau }_r^2+{\tau }_{\varphi }^2}}{{\rho }_f{(r\mathrm{\Omega })}^2}.$$

The local skim friction in non-dimensional form is written as

$$\sqrt{R{e}_r}{C}_f=\sqrt{{[F″\left(0\right)]}^2+{[G\prime \left(0\right)]}^2}.$$

The required torque to rotate a disk of radius $r_0$ , can be determined as

$$T=-\underset{0}{\overset{r_0}{\int }}{\mu }_f{(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r})}_{z=0}2\pi r^2\mathrm{dr}=-\frac{\pi {\rho }_f\mathrm{\Omega }{r}_0^4}{2}\sqrt{2{\upsilon }_f\mathrm{\Omega }}G\prime (0).$$

The rate of heat transfer, can be determined as

$$N{u}_r=\frac{\mathrm{rq}″}{k_f(T_w-T_{\infty })},$$

where, $q″=-{(k_f+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}})\frac{\partial T}{\partial z}|}_{z=0}$ is wall heat flux. Equation (25) gives

$$N{u}_r=-2\sqrt{R{e}_r}(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4}{3}\mathrm{Rd})\theta \prime \left(0\right).$$

Solution methodology

The governing PDEs are transformed to ODEs with the introduction of appropriate similarity quantities. The resulting ODEs given in equations (9)−(11) along-with BCs given in equation (12), are solved numerically with MATLAB bvp-4c solver. We make use of following substitutions,

$$y_1=F,y_2=F\prime ,y_3=F″,y_3\prime =F″\prime ,y_4=G,y_5=G\prime ,y_5\prime =G″,$$

$$y_6=\theta ,y_7=\theta \prime ,y_7\prime =\theta ″.$$

The higher order equations are reduced to first order equations along with the initial conditions.

$$\begin{array}{l}{y}_{3^{\prime }}=\frac{K}{2}{y}_2-\frac{1}{2{\mathrm{\Omega }}_2}\\ \{2y_1{y}_3-y_2^2+y_4^2+\left(\frac{Gr}{R{e}^2}\right){\mathrm{\Omega }}_3\left(cos\mathrm{\Psi }\right)y_6-{\mathrm{\Omega }}_{1}M(\frac{{\sigma }_{nf}}{{\sigma }_f})y_2-F_r{y}_2^2\}=0,\end{array}$$

$$\begin{array}{c}{y}_5\prime =\frac{K}{2}{y}_4-\frac{1}{2{\mathrm{\Omega }}_2}\hfill \\ \{2y_1{y}_5-y_2{y}_4+\left(\frac{\mathrm{Gr}}{R{e}^2}\right){\mathrm{\Omega }}_3\left(\sin \mathrm{\Psi }\right)y_6-{\mathrm{\Omega }}_{1}M\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)y_6-F_r{y}_4^2\}=0,\hfill \end{array}$$

$$\begin{array}{c}{y}_7\prime =-\frac{\mathrm{Pr}}{{\mathrm{\Omega }}_4(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4}{3}\mathrm{Rd})}[y_1{y}_7+6\left(\frac{\mathrm{Ec}}{\mathrm{Re}}\right){\mathrm{\Omega }}_5{y}_2^2+\mathrm{Ec}{\mathrm{\Omega }}_5(y_3^2+y_5^2)\hfill \\ +\frac{1}{2}{\mathrm{\Omega }}_4\{\mathrm{MEc}\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)(y_2^2+y_4^2)+\frac{1}{{\mathrm{\Omega }}_2\mathrm{Pr}}\left(\frac{k_{\mathrm{nf}}}{k_f}\right)(A^{*}{y}_2+B^{*}{y}_6)\}\hfill \\ +\mathrm{Ec}{\mathrm{\Omega }}_{5}K(y_2^2+y_4^2)+\mathrm{Ec}{\mathrm{\Omega }}_4{F}_r(y_2^3+y_4^3)].\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{y}_2\left(0\right)=\gamma +{\gamma }_1{(1-\xi y_3\left(0\right))}^{\frac{-1}{2}}{y}_3\left(0\right),\hfill \\ \multicolumn{1}{c}{y_4\left(0\right)=1+{\gamma }_1{(1-\xi y_5\left(0\right))}^{\frac{-1}{2}}{y}_5\left(0\right),}\\ \multicolumn{1}{c}{y_1\left(0\right)=0,y_7\left(0\right)=-\left(\frac{k_f}{k_{\mathrm{nf}}}\right)\mathrm{Bi}(1-y_6\left(0\right)),}\\ \multicolumn{1}{c}{y_2\left(\infty \right)=0,y_4\left(\infty \right)=0,y_6\left(\infty \right)=0.}\end{array}}\end{array}$$

Results and discussion

In order to make a strong understanding of the model, the solutions for velocity and temperature profiles are assembled numerically via MATLAB built-in bvp-4c solver and illustrated graphically. The relations for predicting thermo-physical properties of nanofluids are given in Table 1. The numerical values of thermo-physical properties of base lubricant (PAO) and nanoparticles $(A{l}_2{O}_3)$ are given in Table 2. Figure 2 unveils the effects of inertial parameter $F_r$ on $F^{\prime }\left(\eta \right)$ , $G\left(\eta \right)$ , and $\theta \left(\eta \right)$ . The increasing values of $F_r$ decline both the velocity profiles but favor the temperature of the nanolubricant $A{l}_2{O}_3-\mathrm{PAO}$ . Physically, the boundary layer thickness depends upon the inertial parameter $F_r$ , which gets widen with an escalation in $F_r$ . Therefore, the movement of $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant slows down along both the radial and tangential directions, whereas its temperature gets boosted. The behavior of radial velocity $F^{\prime }\left(\eta \right)$ and tangential velocity $\mathrm{G}\left(\eta \right)$ of $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant due to enhancing values of Grashof number $\mathrm{Gr}$ is presented in Figure 3. The growing values of $\mathrm{Gr}$ oppose the flow of nanolubricant along tangential direction but support the radial flow and hence reduction in tangential velocity $G\left(\eta \right)$ is resulted, while converse response occurs for radial velocity $F^{\prime }\left(\eta \right).$ Figure 4 displays the impacts of velocity slip parameter ${\gamma }_1$ on tangential and radial velocities of $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant. Both the velocity profiles of $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant get decreased by increasing ${\gamma }_1$ . It is also perceived that when ${\gamma }_1\to \infty ,$ the rotating disk stops rotating the nanolubricant. The flow becomes entirely potential for higher values of ${\gamma }_1$ due to which motion of the nanolubricant is resisted. The effects of magnetic parameter $M$ on $F^{\prime }\left(\eta \right)$ , $G\left(\eta \right)$ , and $\theta \left(\eta \right)$ have been portrayed in Figure 5. The enhancing values of $M$ give rise to the Lorentz force which causes resistance to the flow of nanolubricant due to which the tangential and radial velocities of the nanolubricant decrease but its temperature goes up significantly. The imposed magnetic field plays a major role in adjusting and controlling movement of the nanolubricant over surface of the disk. Figures 6 and 7 unveil the influences of non-uniform parameters $A^{*}$ and $B^{*}$ respectively on temperature profile $\theta \left(\eta \right)$ . When we enhance $A^{*}$ and $B^{*}$ positively they are known as heat sources and act as heat generators. They serve as releasing agents of heat energy into fluid flow and hence temperature of the nanolubricant gets improved. Additionally, when $A^{*}$ and $B^{*}$ are given negative values they are known to be heat sinks. They act as heat absorbers and cause a significant drop in temperature of the nanolubricant. Moreover, heat source or sink can be implemented as a physical tool to achieve desired cooling or heating rate in industrial sector and heat exchangers. The variation in $\theta \left(\eta \right)$ corresponding to Radiation parameter $\mathrm{Rd}$ is visualized in Figure 8. The thermal boundary gets enhanced by enhancing $\mathrm{Rd}$ due to which temperature of the nanolubricant rises. Figure 9 exposes the effects of Biot number $\mathrm{Bi}$ on temperature profile $\theta \left(\eta \right).$ The observation reveals that temperature of the nanolubricant falls by increasing $\mathrm{Bi}$ . Physically, in present case the coefficient of heat transfer gets decreased by enhancing $\mathrm{Bi}$ , which caused a decline in temperature of the nanolubricant. Table 3 presents the comparison of present computational results with the published work. It is pertinent to observe that present results excellently agree with the literature. Table 4 elaborates the effects of different parameters on Nusselt number. It is found that the growing values of radiation parameter and non-uniform heat generation parameters tend to increase value of Nusselt number. The value of Nusselt number drops with rising values of Biot number and non-uniform heat sink parameters.

Table 1.

Thermo-physical aspects of nanolubricant.

Properties	Nanofluid
Dynamic viscosity	$\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}={(1-\varphi )}^{-2.5}$
Density	$\frac{{\rho }_{\mathrm{nf}}}{{\rho }_f}=\{\varphi \frac{{\rho }_s}{{\rho }_f}+(1-\varphi )\}$
Heat capacitance	$\frac{{\left(\rho c_p\right)}_{\mathrm{nf}}}{{\left(\rho c_p\right)}_f}=\{\varphi \frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}+(1-\varphi )\}$
Thermal expansion coefficient	$\frac{{(\rho \beta )}_{\mathrm{nf}}}{{(\rho \beta )}_f}=\{\varphi \frac{{(\rho \beta )}_s}{{(\rho \beta )}_f}+(1-\varphi )\}$
Electrical conductivity	$\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=(1-\varphi )+\varphi \frac{{\sigma }_s}{{\sigma }_f}$
Thermal conductivity	$\frac{k_{\mathrm{nf}}}{k_f}=\frac{((s-1)k_f+k_s)-(s-1)\varphi (k_f-k_s)}{((s-1)k_f+k_s)+\varphi (k_f-k_s)}$

Table 2.

The features of $\mathrm{PAO}$ (base fluid) and $A{l}_2{O}_3$ (nanoparticles). ⁴⁴

S. no.		$A{l}_2{O}_3$	$\mathrm{PAO}$
$1$	$\rho (\mathrm{Kg}/m^3)$	$3970$	$798$
$2$	$C_p(J/\mathrm{KgK})$	$765$	$2303$
$3$	$k(W/\mathrm{mK})$	$40$	$0.143$
$4$	$\sigma (S/m)$	$1.0\times {10}^{-5}$	$1.0\times {10}^{-11}$
5	$\beta (1/K)$	$0.85\times {10}^{-5}$	$3.5\times {10}^{-4}$

Figure 2.

Effects of $F_r$ on $F\prime (\eta )$ , $G\left(\eta \right)$ , and $\theta \left(\eta \right).$

Figure 3.

Effects of $\mathrm{Gr}$ on $F\prime (\eta )$ and $G\left(\eta \right).$

Figure 4.

Effects of ${\gamma }_1$ on $F\prime (\eta )$ and $G\left(\eta \right).$

Figure 5.

Effects of $M$ on $F\prime (\eta )$ , $G\left(\eta \right)$ , and $\theta \left(\eta \right).$

Figure 6.

Impact of $A^{*}$ on $\theta \left(\eta \right).$

Figure 7.

Impact of $B^{*}$ on $\theta \left(\eta \right).$

Figure 8.

Impact of $\mathrm{Rd}$ on $\theta \left(\eta \right).$

Figure 9.

Impact of $\mathrm{Bi}$ on $\theta \left(\eta \right).$

Table 3.

The comparison of present computational results of Skin friction with Nayak et al., ¹⁶ while additional assumptions are omitted.

$\mathrm{\Gamma }$	Skin friction
	$M=1.0$		$M=3.0$		$M=5.0$
	Nayak et al. 16	Present work	Nayak et al. 16	Present work	Nayak et al. 16	Present work
$0.1$	$1.46870$	$1.468704$	$1.89639$	$1.896392$	$2.21627$	$2.216271$
$0.5$	$0.94252$	$0.94252$ 3	$1.12955$	$1.12955$ 4	$1.24770$	$1.24770$ 2
$0.8$	$0.74860$	$0.748601$	$0.87208$	$0.872083$	$0.94436$	$0.944360$

Table 4.

Effects of different parameters on Nusselt number.

$\mathrm{Pr}$	$\mathrm{Ec}$	$\mathrm{Rd}$	$A^{*}$	$B^{*}$	$\mathrm{Bi}$	$\mathrm{N}{\mathrm{u}}_r$
$6\mathbf{.}1$						$1.1318$
$6\mathbf{.}3$	$0.4$	$2.2$	$0.4$	$0.3$	$0.8$	$1.0296$
$6\mathbf{.}5$						$0.9441$
	$0\mathbf{.}3$					$1.2691$
$6.3$	$0\mathbf{.}5$	$2.2$	$0.4$	$0.3$	$0.8$	$2.0559$
	$0\mathbf{.}7$					$3.3642$
		$2\mathbf{.}1$				$0.9601$
$6.3$	$0.3$	$2\mathbf{.}3$	$0.4$	$0.3$	$0.8$	$1.1633$
		$2\mathbf{.}5$				$1.3788$
			$1\mathbf{.}0$			$1.4271$
$6.3$	$0.3$	$2.2$	$2\mathbf{.}0$	$0.3$	$0.8$	$1.5381$
			$3\mathbf{.}0$			$1.6128$
				$\mathbf{-}0\mathbf{.}1$		$0.8741$
$6.3$	$0.3$	$2.2$	$0.4$	$\mathbf{-}0\mathbf{.}5$	$0.8$	$0.8221$
				$\mathbf{-}0\mathbf{.}9$		$0.7743$
					$0\mathbf{.}4$	$0.8616$
$6.3$	$0.3$	$2.2$	$0.4$	$0.3$	$0\mathbf{.}6$	$0.8348$
					$0\mathbf{.}8$	$0.8119$

Concluding remarks

In this work, an attempt is made to develop a mathematical model to discuss the flow of magnetized $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant over an inclined rotating disk with Thompson and Troian slip at the boundary. The influences of mixed convection, nonlinear heat radiation, viscous dissipation, Joule heating, and non-uniform heat sink/source are also included in the modeling. The flow is supposed to take place in a Darcy-Forchheimer porous medium. We have solved the proposed model numerically with MATLAB built-in bvp-4c method after metamorphosing the PDEs to ODEs. The key findings of this work are as follows:

• The enhancing values of velocity slip parameter decrease the tangential and radial velocities of the nanolubricant $A{l}_2{O}_3-\mathrm{PAO}.$

• The growing values of Grashof number oppose the flow of nanolubricant along tangential direction but support the radial flow.

• With an escalation in inertial parameter, the movement of $A{l}_2{O}_3-\mathrm{PAO}$ nanolubricant slows down along both the radial and tangential directions, whereas its temperature gets boosted.

• The temperature of the nanolubricant $A{l}_2{O}_3-\mathrm{PAO}$ enhances significantly by strengthening the magnetic field, whereas radial and tangential velocities get retarded.

• The non-uniform heat source/sink parameters play a vital role in controlling heat transmission phenomenon.

• The temperature of the nanolubricant $A{l}_2{O}_3-\mathrm{PAO}$ falls by increasing Biot number and converse behavior is observed for radiation parameter.

• The growing values of Eckert number, radiation parameter, and non-uniform heat source parameters tend to increase the value of Nusselt number.

• The value of Nusselt number drops with rising values of Biot number and non-uniform heat sink parameters.